Integration by Parts

Substitution handles the chain rule in reverse. The integration by parts (IBP) formula handles the product rule in reverse: if $F(x) = u(x)\cdot v(x)$, then $(uv)' = u'v + uv'$, so $uv' = (uv)' - u'v$, and integrating both sides gives $\int u\,dv = uv - \int v\,du$. This formula is the core tool for integrands that are products of functions from different families — $\int xe^x\,dx$, $\int x\sin x\,dx$, $\int \ln x\,dx$ — and it underlies the derivation of probability-weighted option payoffs (where the payoff function multiplied by the density is the relevant product).

Learning objectives. By the end of this note you should be able to: